On the existence of logarithmic and orbifold jet differentials

We introduce the concept of directed orbifold, namely triples (X, V, D) formed by a directed algebraic or analytic variety (X, V), and a ramification divisor D, where V is a coherent subsheaf of the tangent bundle TX. In this context, we introduce an algebra of orbifold jet differentials and their sections. These jet sections can be seen as algebraic differential operators acting on germs of curves, with meromorphic coefficients, whose poles are supported by D and multiplicities are bounded by the ramification indices of the components of D. We estimate precisely the curvature tensor of the corresponding directed structure V[D] in the general orbifold case-with a special attention to the compact case D = 0 and to the logarithmic situation where the ramification indices are infinite. Using holomorphic Morse inequalities on the tautological line bundle of the projectivized orbifold Green-Griffiths bundle, we finally obtain effective sufficient conditions for the existence of global orbifold jet differentials.


Introduction and main definitions
The present work is concerned primarily with the existence of logarithmic and orbifold jet differentials on projective varieties. For the sake of generality, and in view of potential applications to the case of foliations, we work throughout this paper in the category of directed varieties, and generalize them by introducing the concept of directed orbifold. 0.1. Definition. Let X be a complex manifold or variety. A directed structure (X, V ) on X is defined to be a subsheaf V ⊂ O(T X ) such that O(T X )/V is torsion free. A morphism of directed varieties Ψ : (X, V ) → (Y, W ) is a holomorphic map Ψ : X → Y such that dΨ(V ) ⊂ Ψ * W . We say that (X, V ) is non singular if X is non singular and V is locally free, i.e., is a holomorphic subbundle of T X .
We refer to the absolute case as being the situation when V = T X , the relative case when V = T X/S for some fibration X → S, and the foliated case when V is integrable, i.e. [V, V ] ⊂ V , that is, V is the tangent sheaf to a holomorphic foliation. We now combine these concepts with orbifold structures in the sense of Campana [Cam04]. 0.2. Definition. A directed orbifold is a triple (X, V, D) where (X, V ) is a directed variety and D = (1 − 1 ρ j )∆ j an effective real divisor, where ∆ j is an irreducible hypersurface and ρ j ∈ ]1, ∞] an associated "ramification number". We denote by ⌈D⌉ = ∆ j the corresponding reduced divisor, and by |D| = ∆ j its support. (a) We will say that (X, V, D) is non singular if (X, V ) is non singular and D is a simple normal crossing divisor such that D is transverse to V . If r = rank(V ), we mean by this that there are at most r components ∆ j meeting at any point x ∈ X, and that for any p-tuple (j 1 , . . . , j p ) of indices, 1 p r, we have dim V x ∩ p j=1 T ∆ j ℓ ,x = r − p at any point x ∈ p j=1 ∆ j ℓ .
(b) If (X, V, D) is non singular, the canonical divisor of (X, V, D) is defined to be K V,D = K V + D (in additive notation), where K V = det V * . (c) The so called logarithmic case corresponds to all multiplicities ρ j = ∞ being taken infinite, so that D = ∆ j = ⌈D⌉.
In case V = T X , we recover the concept of orbifold introduced in [Cam04], except possibly for the fact that we allow here ρ j > 1 to be real or ∞, (even though the case where ρ j is in N ∪ {∞} is of greater interest). It would certainly be interesting to investigate the case when (X, V, D) is singular, by allowing singularities in V and tangencies between V and D, and to study whether the results discussed in this paper can be extended in some way, e.g. by introducing suitable multiplier ideal sheaves taking care of singularities, as was done in [Dem15] for the study of directed varieties (X, V ). For the sake of technical simplicity, we will refrain to do so here, and will therefore leave for future work the study of singular directed orbifolds. 0.3. Definition. Let (X, V, D) be a singular directed orbifold. We say that f : C → X is an orbifold entire curve if f is a non constant holomorphic map such that : (a) f is tangent to V (i.e. f ′ (t) ∈ V f (t) at every point, or equivalently f : (C, T C ) → (X, V ) is a morphism of directed varieties ; (b) f (C) is not identically contained in |D| ; (c) at every point t 0 ∈ C such that f (t 0 ) ∈ ∆ j , f meets ∆ j with ramification number ρ j , i.e., if ∆ j = {z j = 0} near f (t 0 ), then z j • f (t) vanishes with multiplicity ρ j at t 0 . In the case of a logarithmic component ∆ j (ρ j = ∞), condition (c) is to be replaced by the assumption (c ′ )f (C) does not meet ∆ j .
One can now consider a category of directed orbifolds as follows.
It is easy to check that, if the image of the composed morphism is not contained in the support of the divisor on the target space, the composite of directed orbifold morphisms is actually a directed orbifold morphism, and that the composition of an orbifold entire curve f : C → (X, V, D) with a directed orbifold morphism Ψ : (X, V, D) → (Y, W, D ′ ) produces an orbifold entire curve Ψ • f : C → (Y, W, D ′ ) (provided that Ψ • f (C) ⊂ |D ′ |). One of our main goals is to investigate the following orbifold generalization of the Green-Griffiths conjecture. 0.5. Conjecture. Let (X, V, D) be a non singular directed orbifold of general type, in the sense that the canonical divisor K V + D is big. Then then exists an algebraic subvariety Y X containing all orbifold entire curves f : C → (X, V, D).
As in the absolute case (V = T X , D = 0), the idea is to show, at least as a first step towards the conjecture, that orbifold entire curves must satisfy suitable algebraic differential equations. In section 1, we introduce graded algebras (0.6) m∈N E k,m V * D of sheaves of "orbifold jet differentials". These sheaves correspond to algebraic differential operators P (f ; f ′ , f ′′ , . . . , f (k) ) acting on germs of k-jets of curves that are tangent to V and satisfy the ramification conditions prescribed by D. The strategy relies on the following orbifold version of the vanishing theorem, whose proof is sketched in the appendix. 0.7. Proposition. Let (X, V, D) be a projective non singular directed orbifold, and A an ample divisor on X. Then, for every orbifold entire curve f : C → (X, V, D) and every global jet differential operator P ∈ H 0 (X, E k,m V * D ⊗ O X (−A)), we have P (f ; f ′ , f ′′ , . . . , f (k) ) = 0. The next step consists precisely of finding sufficient conditions that ensure the existence of global sections P ∈ H 0 (X, E k,m V * D ⊗ O X (−A)). Recall that it has been shown in [CDR20,Proposition 5.1] that the general type assumption is not a sufficient condition for the existence of global jet differentials. Among more general results, we obtain 0.8. Theorem. Let D = j (1 − 1 ρ j )∆ j a simple normal crossing orbifold divisor on P n with deg ∆ j = d j . Then there exist non zero jet differentials of order k and large degree m on P n D , with a small negative twist O P n (−mτ ), τ > 0, under any of the following two sufficient conditions : ∼ (2π) 1/2 n n+7/2 e −n (γ + log n) n−1 .
Let us recall some related results previously obtained in this orbifold setting. In the case of orbifold surfaces P 2 , 1 − 1 ρ C where C is a smooth curve of degree d, such existence results have been obtained in [CDR20] for k = 2, d 12 and ρ 5 depending on d. In [DR20], the existence of jet differentials is obtained for orbifolds P n , d i=1 1 − 1 ρ H i in any dimension for k = 1, ρ 3 along an arrangement of hyperplanes of degree d 2n 2n ρ−2 + 1 . In [BD18], it is established that the orbifold P n , 1 − 1 d D , where D is a general smooth hypersurface of degree d, is hyperbolic i.e. there is no non-constant orbifold entire curve f : C → P n , 1 − 1 d D , if d (n + 2) n+3 (n + 1) n+3 .
The proof of Theorem 0.8 depends on a number of ingredients and on rather extensive curvature calculations. The first point is that the curvature tensor of the orbifold directed structure V D can be controlled in a precise manner. This is detailed in § 6.A. 0.9. Theorem. Assume that X is projective. Given an an ample line bundle A on X, let γ V be the infimum of real numbers γ 0 such that γ Θ A ⊗ Id V − Θ V is positive in the sense of Griffiths, for suitable C ∞ smooth hermitian metrics on V . Assume that D = j (1 − 1/ρ j )∆ j is transverse to V , and select d j 0 such that d j A − ∆ j is nef. Then for γ > γ V,D := max(max(d j /ρ j ), γ V ) 0 and for suitable hermitian metrics on A, V , O X (∆ j ), the "orbifold metric" yields a curvature tensor γ Θ A ⊗ Id − Θ V D such that the associated quadratic form Q V D ,γ,ε on T X ⊗ V satisfies for ε N ≪ ε N −1 ≪ · · · ≪ ε 1 ≪ 1 the curvature estimate Here, the symbol ≃ means that the ratio of the left and right hand sides can be chosen in [1−α, 1+α] for any α > 0 prescribed in advance.
The next argument is the observation that the sheaf O X (E k,m V * D ) is the direct image of a certain tautological rank 1 sheaf O X k (V D ) (m) on the "orbifold k-jet bundle" X k (V D ) → X. Choosing hermitian metrics according to Theorem 0.9, one then gets a hermitian metric on O X k (V D ) (1) associated with an "orbifold Finsler metric" on the bundle J k V of k-jets of holomorphic curves f : (C, 0) → (X, V ). In normalized coordinates (z 1 , . . . , z n ) on X and on V , the latter can be expressed as , f ∈ J k V, f (0) = x, at any point x ∈ X where ∆ j = {z j = 0}, 1 j p, r = rank V . An application of holomorphic Morse inequalities ([Dem85], see also § 2, 3, 4) then provides asymptotic estimates of the dimensions of the cohomology groups ). This is done in several steps. Section § 4 expresses the Morse integrals that need to be computed. Section § 5 establishes some general estimates of Chern forms related to the curvature tensor Θ E,h of a given hermitian vector bundle (E, h), under suitable positivity assumptions. More precisely, Proposition 5.13 gives upper and lower bounds of integrals of the form (0.12) in terms of Tr E Θ E,h = Θ det E,det h , where µ is the unitary invariant probability measure on the unit sphere bundle S(E), and the ℓ j are linear forms. As far as we know, these estimates seem to be new. Sections § 6.B and § 7 then proceed with the detailed calculations of the orbifold and logarithmic Morse integrals involved in the problem. It is remarkable that a large part of the calculations use Chern forms and are non cohomological, although the final bounds are purely cohomological. At this point, we do not have a complete explanation of this "transcendental" phenomenon.

Logarithmic and orbifold jet differentials 1.A. Directed varieties and associated jet differentials
Let (X, V ) be a non singular directed variety. We set n = dim C X, r = rank C V , and following the exposition of [Dem97], we denote by π k : J k V → X the bundle of k-jets of holomorphic curves tangent to V at each point. The canonical bundle of V is defined to be is a germ of holomorphic curve tangent to V , we denote by f [k] (0) its k-jet at t = 0. For x 0 ∈ X given, we take a coordinate system (z 1 , . . . , z n ) centered at x 0 such that V x 0 = Span( ∂ ∂zµ ) 1 µ r . Then there exists a neighborhood U of x 0 such that V |U admits a holomorphic frame (e µ ) 1 µ r of the form with a λµ (0) = 0. Germs of curves f : (C, 0) → X tangent to V |U are obtained by integrating the system of ordinary differential equations when we write f = (f 1 , . . . , f n ) in coordinates. Therefore any such germ of curve f is uniquely determined by its initial point z = f (0) and its projectionf = (f 1 , . . . , f r ) on the first r coordinates.
is uniquely determined by its initial point f (0) = z ≃ (z 1 , . . . , z n ) and the Taylor expansion of order k Alternatively, we can pick an arbitrary local holomorphic connection ∇ on V |U and represent the This gives a local biholomorphic trivialization of J k V |U of the form . . , ξ k ) = (∇f (0), . . . , ∇f k (0)) ; the particular choice of the "trivial connection" ∇ 0 of V |U that turns (e µ ) 1 µ r into a parallel frame precisely yields the components ξ s ∈ V |U ≃ C r appearing in (1.4). We could of course also use a C ∞ connection ∇ = ∇ 0 + Γ where Γ ∈ C ∞ (U, T * X ⊗ Hom(V, V )), and in this case, the corresponding trivialization (1.5) is just a C ∞ diffeomorphism; the advantage, though, is that we can always produce such a global C ∞ connection ∇ by using a partition of unity on X, and then (1.5) becomes a global C ∞ diffeomorphism. Now, there is a global holomorphic C * action on J k V given at the level of germs by f → α · f where α · f (t) := f (αt), α ∈ C * . With respect to our trivializations (1.5), this is the weighted C * action defined by (1.6) α · (ξ 1 , ξ 2 , . . . , ξ k ) = (αξ 1 , α 2 ξ 2 , . . . , α k ξ k ), ξ s ∈ V.
We see that J k V → X is an algebraic fiber bundle with typical fiber C rk , and that the projectivized k-jet bundle , . . . , k [r] ) weighted projective bundle over X, of total dimension (1.8) dim X k (V ) = n + kr − 1.
1.9. Definition. We define O X (E k,m V * ) to be the sheaf over X of holomorphic functions P (z ; ξ 1 , . . . , ξ k ) on J k V that are weighted polynomials of degree m in (ξ 1 , . . . , ξ m ).
In coordinates and in multi-index notation, we can write where the a α 1 ...α k (z) are holomorphic functions in z = (z 1 , . . . , z n ) and ξ αs s actually means ξ αs s = ξ α s,1 s,1 . . . ξ αs,r s,r for ξ s = (ξ s,1 , . . . , ξ s,r ) ∈ C r , α s = (α s,1 , . . . , α s,r ) ∈ N r , and |α s | = r j=1 α s,j . Such sections can be interpreted as algebraic differential operators acting on holomorphic curves f : Here f (s) (t) αs is actually to be expanded as . . , f (k) ) when we want to make more explicit the dependence of the expression in terms of the derivatives of f . We thus get a sheaf of graded algebras Locally in coordinates, the algebra is isomorphic to the weighted polynomial ring An immediate consequence of these definitions is : 1.13. Proposition. The projectivized bundle π k : X k (V ) → X can be identified with and, if O X k (V ) (m) denote the associated tautological sheaves, we have the direct image formula 1.14. Remark. These objects were denoted X GG k and E GG k,m V * in our previous paper [Dem97], as a reference to the work of Green-Griffiths [GG79], but we will avoid here the superscript GG to simplify the notation.
Thanks to the Faà di Bruno formula, a change of coordinates w = ψ(z) on X leads to a transformation rule where Q ψ is a polynomial of weighted degree k in the lower order derivatives. This shows that the transformation rule of the top derivative is linear and, as a consequence, the partial degree in f (k) of the polynomial P (f ; f ′ , . . . , f k) ) is intrinsically defined. By taking the corresponding filtration and factorizing the monomials (f (k) ) α k with polynomials in f ′ , f ′′ , . . . , f (k−1) , we get graded pieces By considering successively the partial degrees with respect to f (k) , f (k−1) , . . . , f ′′ , f ′ and merging inductively the resulting filtrations, we get a multi-filtration such that

1.B. Logarithmic directed varieties
We now turn ourselves to the logarithmic case. Let (X, V, D) be a non singular logarithmic variety, where D = ∆ j is a simple normal crossing divisor. Fix a point x 0 ∈ X. By the assumption that D is transverse to V , we can then select holomorphic coordinates (z 1 , . . . , z n ) centered at x 0 such that V x 0 = Span( ∂ ∂z j ) 1 j r and ∆ j = {z j = 0}, 1 j p, are the components of D that contain x 0 (here p r and we can have p = 0 if x 0 / ∈ |D|). What we want is to introduce an algebra of differential operators, defined locally near x 0 as the weighted polynomial ring For this we notice that . A similar argument easily shows that the above graded rings do not depend on the particular choice of coordinates made, as soon as they satisfy ∆ j = {z j = 0}. Now (as is well known in the absolute case V = T X ), we have a corresponding logarithmic directed structure V D and its dual V * D . If the coordinates (z 1 , . . . , z n ) are chosen so that V x 0 = {dz r+1 = . . . = dz n = 0}, then the fiber V D x 0 is spanned by the derivations , . . . , ∂ ∂z r .
The dual sheaf O X (V * D ) is the locally free sheaf generated by It follows from this that O X (V D ) and O X (V * D ) are locally free sheaves of rank r. By taking det(V * D ) and using the above generators, we find in additive notation. Quite similarly to 1.13 and 1.15, we have : be the graded algebra defined in coordinates by (1.16) or (1.16 ′ ). We define the logarithmic k-jet bundle to be If O X k (V D ) (m) denote the associated tautological sheaves, we get the direct image formula Moreover, the multi-filtration by the partial degrees in the derivatives f (s) j has graded pieces

1.C. Orbifold directed varieties
We finally consider a non singular directed orbifold (X, V, D), where D = (1 − 1 ρ j )∆ j is a simple normal crossing divisor transverse to V . Let ⌈D⌉ = ∆ j be the corresponding reduced divisor. By § 1.B, we have associated logarithmic sheaves O X (E k,m V * ⌈D⌉ ). We want to introduce a graded subalgebra in such a way that for every germ P ∈ O X (E k,m V * D ) and every germ of orbifold curve f : (C, 0) → (X, V, D) the germ of meromorphic function P (f )(t) is bounded at t = 0 (hence holomorphic). Assume that ∆ 1 = {z 1 = 0} and that f has multiplicity q ρ 1 > 1 along ∆ 1 at t = 0. Then f to be bounded is to require that the multiplicities of poles satisfy is taken to be the graded ring generated by monomials (1.20) of degree s|α s | = m, satisfying the pole multiplicity conditions (1.20 ′ ). These conditions do not depend on the choice of coordinates, hence we get a globally and intrinsically defined sheaf of algebras on X.
Proof. We only have to prove the last assertion. Consider a change of variables w = ψ(z) such that ∆ j can still be expressed as ∆ j = {w j = 0}. Then, for j = 1, . . . , p, we can write w j = z j u j (z) with an invertible holomorphic factor u j . We need to check that the monomials (1.20) computed with g = ψ •f are holomorphic combinations of those associated with f . However, we have g j = f j u j (f ), hence g (s) j (u j (f )) (s−ℓ) by the Leibniz formula, and we see that expands as a linear combination of monomials The above conditions (1.20 ′ ) suggest to introduce as in [CDR20] a sequence of "differentiated" orbifold divisors We say that D (s) is the order s orbifold divisor associated to D ; its ramification numbers are ρ (s) j = max(ρ j /s, 1). By definition, the logarithmic components (ρ j = ∞) of D remain logarithmic in D (s) , while all others eventually disappear when s is large. Now, we introduce (in a purely formal way) a sheaf of rings O X = O X [z • j ] by adjoining all positive real powers of coordinates z j such that ∆ j = {z j = 0} is locally a component of D. Locally over X, this can be done by taking the universal cover Y of a punctured polydisk in the local coordinates z j on X. If γ : Y → D * (0, r) ֒→ X is the covering map and U ⊂ D(0, r) is an open subset, we can then consider the functions of O X (U ) as being defined on γ −1 (U ∩ D * (0, r)).
In case X is projective, one can even achieve such a construction "globally", at least on a Zariski open set, by taking Y to be the universal cover of a complement X (|D| ∪ |A|), where A = A j is a very ample normal crossing divisor transverse to D, such that O X (∆ j ) |X |A| is trivial for every j ; then O X is well defined as a genuine sheaf on X |A|.
In this setting, the subalgebra m O X (E k,m V * D ) still has a multi-filtration induced by the one on m O X (E k,m V * ⌈D⌉ ), and by extending the structure sheaf O X into O X , we get an inclusion is the "s-th orbifold (dual) directed structure", generated by the order s differentials (1.24) z −(1−s/ρ j ) + j d (s) z j , 1 j p, d (s) z j , p + 1 j r.
By construction, we have 1.26. Remark. When ρ j = a j /b j ∈ Q + , one can find a finite ramified Galois cover g : Y → X from a smooth projective variety Y onto X, such that the compositions (z j • g) 1/a j become singlevalued functions w j on Y . In this way, the pull-back O Y (g * V * D (s) ) is actually a locally free O Y -module. On can also introduce a sheaf of algebras which we will denote by O Y (E k,m V * D ), generated, according to the notation of § 1.B, by the elements g * (z (1−s/ρ j ) + j d (s) z j ), 1 j p, and g * (d (s) z j ), p + 1 j r. Then, as already shown in [CDR20], there is indeed a multifiltration on O Y (E k,m V * D ) whose graded pieces are However, we will adopt here an alternative viewpoint that avoids the introduction of finite or infinite covers, and suits better our approach. Using the general philosophy of [Laz??], the idea is to consider a "jet orbifold directed structure" X k (V D ) as the underlying "jet logarithmic directed structure" X k (V ⌈D⌉ ), equipped additionally with a submultiplicative sequence of ideal sheaves . These are precisely defined as the base loci ideals of the local sections defined by (1.20) and (1.20 ′ ), seen as sections of the logarithmic tautological sheaves O X k (V ⌈D⌉ ) (m). The corresponding analytic viewpoint is to consider ad hoc singular hermitian metrics on O X k (V ⌈D⌉ ) (1) whose singularities are asymptotically described by the limit of the formal m-th root of J m D , see § 3.B. It then becomes possible to deal without trouble with real coefficients ρ j ∈ ]1, ∞], and since we no longer have to worry about the existence of Galois covers, the projectivity assumption on X can be dropped as well.

Preliminaries on holomorphic Morse inequalities 2.A. Basic results
We first recall the basic results concerning holomorphic Morse inequalities for smooth hermitian line bundles, first proved in [Dem85].
2.1. Theorem. Let X be a compact complex manifolds, E → X a holomorphic vector bundle of rank r, and (L, h) a hermitian line bundle. We denote by Θ L,h = i 2π ∇ 2 h = − i 2π ∂∂ log h the curvature form of (L, h) and introduce the open subsets of X Then, for all q = 0, 1, . . . , n, the dimensions h q (X, E ⊗ L m ) of cohomology groups of the tensor powers E ⊗ L m satisfy the following "Strong Morse inequalities" as m → +∞ : with equality χ(X, E ⊗ L m ) = r m n n! X Θ n L,h + o(m n ) for the Euler characteristic (q = n). As a consequence, one gets upper and lower bounds for all cohomology groups, and especially a very useful criterion for the existence of sections of large multiples of L.
The following simple lemma is the key to derive algebraic Morse inequalities from their analytic form (cf. [Dem94], Theorem 12.3).
2.3. Lemma. Let η = α−β be a difference of semipositive (1, 1)-forms on an n-dimensional complex manifold X, and let 1l η, q be the characteristic function of the open set where η is non degenerate with a number of negative eigenvalues at most equal to q. Then in particular 1l η, 1 η n α n − nα n−1 ∧ β for q = 1.
Proof. Without loss of generality, we can assume α > 0 positive definite, so that α can be taken as the base hermitian metric on X. Let us denote by λ 1 λ 2 . . . λ n 0 the eigenvalues of β with respect to α. The eigenvalues of η = α − β are then given by hence the open set {λ q+1 < 1} coincides with the support of 1l η, q , except that it may also contain a part of the degeneration set η n = 0. On the other hand we have where σ j n (λ) is the j-th elementary symmetric function in the λ j 's. Thus, to prove the lemma, we only have to check that This is easily done by induction on n (just split apart the parameter λ n and write σ j n (λ) = σ j n−1 (λ)+ σ j−1 n−1 (λ) λ n ).
2.4. Corollary. Assume that η = Θ L,h can be expressed as a difference η = α − β of smooth (1, 1)-forms α, β 0. Then we have SM(q) : and in particular, for q = 1, 2.5. Remark. These estimates are consequences of Theorem 2.1 and Lemma 2.3, by taking the integral over X. The estimate for h 0 was stated and studied by Trapani [Tra93]. In the special case α = Θ A,h A > 0, β = Θ B,h B > 0 where A, B are ample line bundles, a direct proof can be obtained by purely algebraic means, via the Riemann-Roch formula. However, we will later have to use Corollary 2.4 in case α and β are not closed, a situation in which no algebraic proof seems to exist.

2.B. Singular holomorphic Morse inequalities
The case of singular hermitian metrics has been considered in Bonavero's PhD thesis [Bon93] and will be important for us. We assume that L is equipped with a singular hermitian metric h = h ∞ e −ϕ with analytic singularities, i.e., h ∞ is a smooth metric, and on an neighborhood V ∋ x 0 of an arbitrary point x 0 ∈ X, the weight ϕ is of the form where g j ∈ O X (V ) and u ∈ C ∞ (V ). We then have Θ L,h = α + i 2π ∂∂ϕ where α = Θ L,h∞ is a smooth closed (1, 1)-form on X. In this situation, the multiplier ideal sheaves play an important role. We define the singularity set of h by Sing(h) = Sing(ϕ) = ϕ −1 (−∞) which, by definition, is an analytic subset of X. The associated q-index sets are (2.8) X(L, h, q) = x ∈ X Sing(h) ; Θ L,h (x) has signature (n − q, q) .
We can then state: 2.9. Theorem ([Bon93]). Morse inequalities still hold in the context of singular hermitian metric with analytic singularities, provided the cohomology groups under consideration are twisted by the appropriate multiplier ideal sheaves, i.e. replaced by H q (X, E ⊗ L m ⊗ I(h m )).
2.10. Remark. The assumption (2.6) guarantees that the measure 1l X Sing(h) (Θ L,h ) n is locally integrable on X, as is easily seen by using the Hironaka desingularization theorem and by taking a log resolution µ : X → X such that µ * (g j ) = (γ) ⊂ O X becomes a principal ideal associated with a simple normal crossing divisor E = div(γ). Then µ * Θ L,h = c[E] + β where β is a smooth closed (1, 1)-form on X, hence It should be observed that the multiplier ideal sheaves I(h m ) and the integral X Sing(h) Θ n L,h only depend on the equivalence class of singularities of h : if we have two metrics with analytic singularities h j = h ∞ e −ϕ j , j = 1, 2, such that ψ = ϕ 2 − ϕ 1 is bounded, then, with the above notation, we have µ * Θ L,h j = c[E] + β j and β 2 = β 1 + i 2π ∂∂ψ, therefore X β n 2 = X β n 1 by Stokes theorem. By using Monge-Ampère operators in the sense of Bedford-Taylor [BT76], it is in fact enough to assume u ∈ L ∞ loc (X) in (2.6), and ψ ∈ L ∞ (X) here. In general, however, the Morse integrals X(L,h j ,q) (−1) q Θ n L,h j , j = 1, 2, will differ.

2.C. Morse inequalities and semi-continuity
Let X → S be a proper and flat morphism of reduced complex spaces, and let (X t ) t∈S be the fibers. Given a sheaf E over X of locally free O X -modules of rank r, inducing on the fibres a family of sheaves (E t → X t ) t∈S , the following semicontinuity property holds ([CRAS]): 2.11. Proposition. For every q 0, the alternate sum is upper semicontinuous with respect to the (analytic) Zariski topology on S. Now, if L → X is an invertible sheaf equipped with a smooth hermitian metric h, and if (h t ) are the fiberwise metrics on the family (L t → X t ) t∈S , we get where δ(t) → 0 as t → 0. In fact, the proof of holomorphic Morse inequalities shows that the inequality holds uniformly on every relatively compact S ′ ⋐ S, with in the right hand side, and t → I(t) is clearly continuous with respect to the ordinary topology. In other words, the Morse integral computed on the central fibers provides uniform upper bounds for cohomology groups of E t ⊗ L ⊗m t when t is close to 0 in ordinary topology (and also, as a consequence, for t in a complement S S m of at most countably many analytic strata S m S).
2.13. Remark. Similar results would hold when h is a singular hermitian metric with analytic singularities on L → X, under the restriction that the families of multiplier ideal sheaves (I(h m t )) t∈S "never jump".

2.D. Case of filtered bundles
Let E → X be a vector bundle over a variety, equipped with a filtration (or multifiltration) F p (E), and let G = G p (E) → X be the graded bundle associated to this filtration.
2.14. Lemma. In the above setting, one has for every q 0 Proof. One possible argument is to use the well known fact that there is a family of filtered bundles The result is then an immediate consequence of the semi-continuity result 2.11. A more direct very elementary argument can be given as follows: by transitivity of inequalities, it is sufficient to prove the result for simple filtrations; then, by induction on the length of filtrations, it is sufficient to prove the result for exact sequences 0 → S → E → Q → 0 of vector bundles on X. Consider the associated (truncated) long exact sequence in cohomology: By the rank theorem of linear algebra, The result follows, since here h j (X, G) = h j (X, Q) + h j (X, S).

2.E. Rees deformation construction (after Cadorel)
In this short paragraph, we outline a nice algebraic interpretation by Benoît Cadorel of certain semi-continuity arguments for cohomology group dimensions that underline the analytic approach of [Dem11, Lemma 2.12 and Prop. 2.13] and [Dem12, Prop. 9.28] (we will anyway explain again its essential points in § 3, since we have to deal here with a more general situation). Recall after [Cad17, Prop. 4.2, Prop. 4.5], that the Rees deformation construction allows one to construct natural deformations of Green-Griffiths jets spaces to weighted projectivized bundles.
Let (X, V, D) be a non singular directed orbifold, and let g : Y → (X, D) be an adapted Galois cover, as briefly described in remark 1.26, see also [CDR18, § 2.1] for more details. We then get a Green-Griffiths jet bundle of graded algebras E k,• V ⋆ D → Y which admits a multifiltration of associated graded algebra where the tilde means taking pull-backs by g * . Applying the Proj functor, one gets a weighted projective bundle: Then, following mutadis mutandus the arguments of Cadorel, one constructs a family Y • the other fibers Y k,t are isomorphic to the singular quotient J k (Y, V , D)/C * for the natural C * -action by homotheties, where J k (Y, V , D) is the affine algebraic bundle associated with the sheaf of algebras, and ( Applying the semicontinuity result of [Dem95], and working with holomorphic inequalities, we obtain a control about dimensions of cohomology spaces of E k,m V * D in terms of dimensions of cohomology spaces of the a priori simpler graded pieces G • E k,m V * D . This reduces the study of higher order jet differentials to sections of the tautological sheaves on the weighted projective space associated with a direct sum combination of symmetric differentials. In particular, we have Especially, for q = 1, we get

Construction of jet metrics and orbifold jet metrics 3.A. Jet metrics and curvature tensor of jet bundles
Let (X, V ) be a non singular directed variety and h a hermitian metric on V . We assume that h is smooth at this point (but will later relax a little bit this assumption and allow certain singularities). Near any given point z 0 ∈ X, we can choose local coordinates z = (z 1 , . . . , z n ) centered at z 0 and a local holomorphic coordinate frame (e λ (z)) 1 λ r of V on an open set U ∋ z 0 , such that for suitable complex coefficients (c ijλµ ). It is a standard fact that such a normalized coordinate system always exists, and that the Chern curvature tensor i 2π ∇ 2 V,h of (V, h) at z 0 is given by Therefore, ( i 2π c ijλµ ) are the coefficients of −Θ V,h . Up to taking the transposed tensor with respect to λ, µ, these coefficients are also the components of the curvature tensor Let us fix an integer b ∈ N * that is a multiple of lcm(1, 2, . . . , k), and positive numbers 1 = ε 1 ≫ ε 2 ≫ · · · ≫ ε k > 0. Following [Dem11], we define a global weighted Finsler metric on J k V by putting is the hermitian metric h of V evaluated on the fiber V z , z = f (0). The function Ψ h,b,ε satisfies the fundamental homogeneity property with respect to the C * action on J k V , in other words, it induces a hermitian metric on the dual L * The curvature of L k is given by Our next goal is to compute precisely the curvature and to apply holomorphic Morse inequalities to L → X k (V ) with the above metric. This might look a priori like an untractable problem, since the definition of Ψ h,b,ε is a rather complicated one, involving the hermitian metric in an intricate manner. However, the "miracle" is that the asymptotic behavior of Ψ h,b,ε as ε s /ε s−1 → 0 is in some sense uniquely defined, and "splits" according to the natural multifiltration on jet differentials (as already hinted in § 2.E). This leads to a computable asymptotic formula, which is moreover simple enough to produce useful results.
3.6. Lemma. Let us consider the global C ∞ bundle isomorphism J k V → V ⊕k associated with an arbitrary global C ∞ connection ∇ on V → X, and let us introduce the rescaling transformation Such a rescaling commutes with the C * -action. Moreover, if p is a multiple of lcm(1, 2, . . . , k) and the ratios ε s /ε s−1 tend to 0 for all s = 2, . . . , k, the rescaled Finsler metric on every compact subset of V ⊕k {0}, uniformly in C ∞ topology, and the limit is independent of the connection ∇. The error is measured by a multiplicative factor 1 ± O(max 2 s k (ε s /ε s−1 ) s ).

and inductively we get
where P (z ; ξ 1 , . . . , ξ s−1 ) is a polynomial with C ∞ coefficients in z ∈ U , which is of weighted homogeneous degree s in (ξ 1 , . . . , ξ s−1 ). In other words, the corresponding isomorphisms J k V ≃ V ⊕k correspond to each other by a C * -homogeneous transformation (ξ 1 , . . . , ξ k ) → ( ξ 1 , . . . , ξ k ) such that ξ s = ξ s + P s (z ; ξ 1 , . . . , ξ s−1 ). Let us introduce the corresponding rescaled components and it is easily seen, as a simple consequence of the mean value inequality | When b/s is an integer, similar bounds hold for all derivatives D β z,ξ ( ξ s,ε 2b/s − ξ s,ε 2b/s ) and the lemma follows. Now, we fix a point z 0 ∈ X, a local holomorphic frame (e λ (z)) 1 λ r satisfying (3.1) on a neighborhood U of z 0 , and the holomorphic connection ∇ on V |U such that ∇e λ = 0. Since the uniform estimates of Lemma 3.6 also apply locally (provided they are applied on a relatively compact open subset U ′ ⋐ U ), we can use the corresponding holomorphic trivialization to make our calculations. We do this in terms of the rescaled components ξ s = ε s s ∇ s f (0). Then, uniformly on compact subsets of J k V |U {0}, we have and the error term remains of the same magnitude when we take any derivative D β z,ξ . By (3.1) we find The question is thus reduced to evaluating the curvature of the weighted Finsler metric on V ⊕k defined by We set |ξ s | 2 = λ |ξ s,λ | 2 . A straightforward calculation yields the Taylor expansion log Ψ(z ; ξ 1 , . . . , ξ k ) By (3.5), the curvature form of L k = O X k (V ) (1) is given at the central point z 0 by the formula ) of X k (V ) → X can be represented as a quotient of the "weighted ellipsoid" k s=1 |ξ s | 2b/s = 1 by the S 1 -action induced by the weighted C * -action. This suggests to make use of polar coordinates and to set The Morse integrals will then have to be computed for ( 3.9. Proposition. With respect to the rescaled components ξ s = ε s s ∇ s f (0) at z = f (0) ∈ X and the above choice of coordinates (3.8 * ), the curvature of the tautological sheaf L k = O X k (V ) (1) admits an approximate expression Here ( i 2π c ijλµ ) are the coefficients of −Θ V,h , and the error terms admit an upper bound Proof. The error terms on Θ L k come from the differentiation of the error terms on the Finsler metric, found in Lemma 3.6. They can indeed be differentiated if b is a multiple of lcm(1, 2, . . . , k), since 2b/s is then an even integer.
For the calculation of Morse integrals, it is useful to find the expression of the volume form ω kr−1 are probability measures on ∆ / k−1 and (S 2r−1 ) k respectively (µ being the rotation invariant one).
(b) We have the equality

3.B. Logarithmic and orbifold jet metrics
Consider now an arbifold directed structure (X, V, D), where V ⊂ T X is a subbundle, r = rank(V ), and D = (1 − 1 ρ j )∆ j is a normal crossing divisor that is assumed to intersect V transversally everywhere. One then performs very similar calculations to what we did in § 3.A, but with adapted Finsler metrics. Fix a point z 0 at which p components ∆ j meet, and use coordinates (z 1 , . . . , z n ) such that V z 0 is spanned by ( ∂ ∂z 1 , . . . , ∂ ∂zr ) and ∆ j is defined by z j = 0, 1 j p r.
The logarithmic jet differentials are just polynomials in and the corresponding (ε 1 , . . . , ε k )-rescaled Finsler metric is Alternatively, we could replace |f j | −2 |f (s) j | 2 by |(log f j ) (s) | 2 which has the same leading term and differs by a weighted degree s polynomial in the f −1 j f (ℓ) j , ℓ < s ; an argument very similar to the one used in the proof of Lemma 3.6 then shows that the difference is negligible when ε 1 ≫ ε 2 ≫ · · · ≫ ε k . However (3.11) is just the case of the model metric, in fact we get r-tuples ξ s = (ξ s,j ) 1 j r of components produced by the trivialization of the logarithmic bundle O(V D ), such that for 1 s p and ξ s,j = f (s) j for p + 1 s r.
In general, we are led to consider Finsler metrics of the form where h(z) is a variable hermitian metric on the logarithmic bundle V D . In the orbifold case, the appropriate "model" Finsler metric is As a consequence of Remark 2.10, we would get a metric with equivalent singularities on the dual j | 2 are anyway controlled by the "orbifold part" of the summation. Of course, we need to find a suitable Finsler metric that is globally defined on X. This can be done by taking smooth metrics h V,s on V and h j on O X (∆ j ) respectively, as well as smooth connections ∇ and ∇ j . One can then consider the globally defined metric ) are the tautological sections; here, we want the flexibility of not necessarily taking the same hermitian metrics on V to evaluate the various norms ∇ (s) f h V,s . We obtain Finsler metrics with equivalent singularities by just changing the h V,s and h j (and keeping ∇, ∇ j unchanged). If we also change the connections, then an argument very similar to the one used in the proof of Lemma 3.6 shows that the ratio of the corresponding metrics is 1 ± O(max(ε s /ε s−1 )), and therefore arbitrary close to 1 whenever ε 1 ≫ ε 2 ≫ · · · ≫ ε k ; in any case, we get metrics with equivalent singularities. Fix z 0 ∈ X and use coordinates (z 1 , . . . , z n ) as described at the beginning of § 3.B, so that σ j (z) = z j , 1 j p, in a suitable trivialization of O X (∆ j ). Let f be a k-jet of curve such that f (0) = z ∈ X |D| is in a sufficiently small neighborhood of z 0 . By employing the trivial connections associated with the above coordinates, the derivative f (s) is described by components and ξ orb s,j = ξ log s,j = ξ s,j for p + 1 j r. Here ξ orb s,j are to be thought of as the components of f (s) in the "virtual" vector bundle V D (s) , and the fact that the argument of these complex numbers is not uniquely defined is irrelevant, because the only thing we need to compute the norms is |ξ orb s,j |.
and define the orbifold hermitian norm on V D (s) associated with h V,s and h j by v orb With this notation, the orbifold Finsler metric (3.15) on k-jets is reduced to an expression formally identical to what we had in the compact or logarithmic cases. If v is a local holomorphic section of O X (V ), formula (3.16) shows that the norm v orb hs can take infinite values when z ∈ |D|, while, by (3.16 ′ ), the norm is always bounded (but slightly degenerate along |D|) if v is a section of the logarithmic sheaf O X (V ⌈D⌉ ); we think intuitively of the orbifold total space V D (s) as the subspace of V in which the tubular neighborhoods of the zero section are defined by v orb hs < ε for ε > 0.
3.18. Remark. When ρ j ∈ Q, we can take an adapted Galois cover g : However, as already stressed in Remark 1.26, this viewpoint is not needed in our analytic approach.

3.C. Orbifold tautological sheaves and their curvature
In this context, we define the orbifold tautological sheaves to be the logarithmic tautological sheaves O X k (V ⌈D⌉ ) (m) twisted by the multiplier ideal sheaves associated with the dual metric Ψ * k,b,ε (cf. (3.17)), when these are viewed as singular hermitian metrics over the logarithmic k-jet bundle X k (V ⌈D⌉ ). In accordance with this viewpoint, we simply define the orbifold k-jet bundle to be X k (V D ) = X k (V ⌈D⌉ ). The calculation of the curvature tensor is formally the same as in the case D = 0, and we obtain : Here ( i 2π c (s) ijλµ ) are the coefficients of the curvature tensor −Θ V D (s) , hs , and the error terms are O(max 2 s k (ε s /ε s−1 ) s ), uniformly on the projectivized orbifold variety X k (V D ).
Notice, as is clear from the expressions (3.16 ′′ ), (3.17) and the fact that v j = z j v orb j , that our orbifold Finsler metrics always have fiberwise positive curvature, equal to ω k,r,b (ξ), along the fibers of X k (V D ) → X (even after taking into account the so-called error terms, because fiberwise, the functions under consideration are just sums of even powers | ξ orb s | 2b/s in suitable k-jet components, and are therefore plurisubharmonic.)

Existence theorems for jet differentials 4.A. Expression of the Morse integral
Thanks to the uniform approximation provided by Proposition 3.20, we can (and will) neglect the O(ε s /ε s−1 ) error terms in our calculations. Since ω r,k,b is positive definite on the fibers of X k (V D ) → X (at least outside of the axes ξ s = 0), the index of the (1, 1) curvature form where 1l g V,D,k ,q (z, x, u) is the characteristic function of the open set of points where g V,D,k (z, x, u) has signature (n − q, q) in terms of the dz j 's. Notice that since g V,D,k (z, x, u) n is a determinant, the product 1l g V,D,k ,q (z, x, u) g V,D,k (z, x, u) n gives rise to a continuous function on X k (V D ). By Formula 3.10 (b), we get We assume here that we are either in the "compact" case (D = 0), or in the logarithmic case (ρ j = ∞). Then the curvature coefficients c (s) ijλµ = c ijλµ do not depend on s and are those of the dual bundle V * (resp. V * D ). In this situation, formula 3.20 (b) for g V,D,k (z, x, u) can be thought of as a "Monte Carlo" evaluation of the curvature tensor, obtained by averaging the curvature at random points u s ∈ S 2r−1 with certain positive weights x s /s ; we then think of the k-jet f as some sort of random variable such that the derivatives ∇ k f (0) (resp. logarithmic derivatives) are uniformly distributed in all directions. Let us compute the expected value of (x, u) → g V,D,k (z, x, u) with respect to the probability measure dν k,r (x) dµ(u). Since S 2r−1 u s,λ u s,µ dµ(u s ) = 1 r δ λµ and we find the expected value In other words, we get the normalized trace of the curvature, i.e.
(4.3) E(g V,D,k (z, •, •)) = 1 kr 1 + 1 2 where Θ det(V * D ),det h * is the (1, 1)-curvature form of det(V * D ) with the metric induced by h. It is natural to guess that g V,D,k (z, x, u) behaves asymptotically as its expected value E(g V,D,k (z, •, •)) when k tends to infinity. If we replace brutally g V,D,k by its expected value in (4.2), we get the integral (n + kr − 1)! n! k! r (kr − 1)! 1 (kr) n 1 + 1 2 + · · · + 1 k n X 1l η,q η n , where η := Θ det(V * D ),det h * and 1l η,q is the characteristic function of its q-index set in X. The leading constant is equivalent to (log k) n /n! k! r modulo a multiplicative factor 1 + O(1/ log k). By working out a more precise analysis of the deviation, the following result has been proved in [Dem11] in the compact case; the more general logarithmic case can be treated without any change, so we state the result in this situation by just transposing the results of [Dem11].
λ ⊗ e µ the curvature tensor of V D with respect to an h-orthonormal frame (e λ ), and put Finally consider the k-jet line bundle . When k tends to infinity, the integral of the top power of the curvature of L k on its q-index set X k (V D )(L k , q) is given by for all q = 0, 1, . . . , n, and the error term O((log k) −1 ) can be bounded explicitly in terms of Θ V D , η and ω. Moreover, the left hand side is identically zero for q > n.
The final statement follows from the observation that the curvature of L k is positive along the fibers of X k (V D ) → X, by the plurisubharmonicity of the weight (this is true even when the error terms are taken into account, since they depend only on the base); therefore the q-index sets are empty for q > n. It will be useful to extend the above estimates to the case of sections of (4.5) is an arbitrary Q-line bundle on X and π k : X k (V D ) → X is the natural projection. We assume here that F is also equipped with a smooth hermitian metric h F . In formulas (4.2-4.4), the curvature Θ L F,k of L F,k takes the form and by the same calculations its normalized expected value is Then the variance estimate for g V,D,F,k is the same as the variance estimate for g V,D,k , and the recentered L p bounds are still valid, since our forms are just shifted by adding the constant smooth term Θ F,h F (z). The probabilistic estimate 4.4 is therefore still true in exactly the same form for L F,k , provided we use g V,D,F,k and η F instead of g V,D,k and η. An application of holomorphic Morse inequalities gives the desired cohomology estimates for provided m is sufficiently divisible to give a multiple of F which is a Z-line bundle.
4.8. Theorem. Let (X, V D ) be a non singular logarithmic directed variety, F → X a Q-line bundle, (V D , h) and (F, h F ) smooth hermitian structure on V D and F respectively. We define Then for all q 0 and all m ≫ k ≫ 1 such that m is sufficiently divisible, we have Green and Griffiths [GrGr80] already checked the Riemann-Roch calculation (4.8 c) in the special case D = 0, V = T * X and F = O X . Their proof is much simpler since it relies only on Chern class calculations, but it cannot provide any information on the individual cohomology groups, except in very special cases where vanishing theorems can be applied; in fact in dimension 2, the Euler characteristic satisfies χ = h 0 − h 1 + h 2 h 0 + h 2 , hence it is enough to get the vanishing of the top cohomology group H 2 to infer h 0 χ ; this works for surfaces by means of a well-known vanishing theorem of Bogomolov which implies in general as soon as K X ⊗ F −1 is big and m ≫ 1. In fact, thanks to Bonavero's singular holomorphic Morse inequalities (Theorem 2.9, cf. [Bon93]), everything works almost unchanged in the case where the metric h on V is taken to a product h = h ∞ e ϕ of a smooth metric h ∞ by the exponential of a quasi-plurisubharmonic weight ϕ with analytic singularities (so that det(h * ) = det(h * ∞ )e −rϕ ). Then η is a (1, 1)-current with logarithmic poles, and we just have to twist our cohomology groups by the appropriate multiplier ideal sheaves I k,m associated with the weight 1 k (1 + 1 2 + · · · + 1 k )m ϕ, since this is the multiple of det V * that occurs in the calculation, up to the factor 1 r × rϕ. The corresponding Morse integrals need only be evaluated in the complement of the poles, i.e., on X(η, q) S where S = Sing(ϕ). Since we still get a lower bound for the H 0 of the latter sheaf (or for the H 0 of the un-twisted line bundle ). If we assume that K V ⊗ F −1 is big, these considerations also allow us to obtain a strong estimate in terms of the volume, by using an approximate Zariski decomposition on a suitable blow-up of X. 4.9. Corollary. If F is an arbitrary Q-line bundle over X, one has , when m ≫ k ≫ 1, in particular there are many sections of the k-jet differentials of degree m twisted by the appropriate power of F if K V ⊗ F −1 is big.
Proof. The volume is computed here as usual, i.e. after performing a suitable modification µ : X → X which converts K V into an invertible sheaf. There is of course nothing to prove if K V ⊗ F −1 is not big, so we can assume Vol(K V ⊗ F −1 ) > 0. Let us fix smooth hermitian metrics h 0 on T X and h F on F . They induce a metric µ * (det h −1 which, by our definition of K V , is a smooth metric. By the result of Fujita [Fuj94] on approximate Zariski decomposition, for every δ > 0, one can find a modification µ δ : X δ → X dominating µ such that If we take a smooth metric h A with positive definite curvature form Θ A,h A , then we get a singular hermitian metric where ϕ is quasi-psh with log poles log |σ E | 2 (mod C ∞ ( X δ )) precisely given by the divisor E. We then only need to take the singular metric h on T X defined by (the choice of the factor 1 r is there to correct adequately the metric on det V ). By construction h induces an admissible metric on V and the resulting curvature current η [E] = current of integration on E. Then the 0-index Morse integral in the complement of the poles is given by and Corollary 4.9 follows from the fact that δ can be taken arbitrary small. 4.10. Remark. Since the probability estimate requires k to be very large, and since all non logarithmic components disappear from D (s) when s is large, the above lower bound does not work in the general orbifold case. In that case, one can only hope to get an interesting result when k is fixed and not too large. This is what we will do in § 6.

5.A. Griffiths, Nakano and strong (semi-)positivity
Let E → X be a holomorphic vector bundle equipped with a hermitian metric. Then E possesses a uniquely defined Chern connection ∇ h compatible with h and such that ∇ 0,1 h = ∂. The curvature tensor of (E, h) is defined to be One can then associate bijectively to Θ E,h a hermitian form Θ E,h on T X ⊗ E, such that and can be written Let (z 1 , . . . , z n ) be a holomorphic coordinate system and (e λ ) 1 λ r a smooth frame of E. If (e λ ) is chosen to be orthonormal, then we can write and more generally Θ E,h (τ, τ ) = 1 2π i,j,λ,µ c ijλµ τ iλ τ jµ for every tensor τ ∈ T X ⊗ E. We now consider three concepts of (semi-)positivity, the first two being very classical. 5.4. Definition. Let θ be a hermitian form on a tensor product T ⊗ E of complex vector spaces. We say that (a) θ is Griffiths semi-positive if θ(ξ ⊗ u, ξ ⊗ u) 0 for every ξ ∈ T and every v ∈ E; (b) θ is Nakano semi-positive if θ(τ, τ ) 0 for every τ ∈ T ⊗ E ; (c) θ is strongly semi-positive if there exist a finite collection of linear forms α j ∈ T * , ψ j ∈ E * such that θ = j |α j ⊗ ψ j | 2 , i.e.
Semi-negativity concepts are introduced in a similar way. (d) We say that the hermitian bundle (E, h) is Griffiths semi-positive, resp. Nakano semi-positive, resp. strongly semi-positive, if Θ E,h (x) ∈ Herm(T X,x ⊗ E x ) satisfies the corresponding property for every point x ∈ X. (e) (Strict) Griffiths positivity means that Θ E,h (ξ ⊗u, ξ ⊗u) > 0 for every non zero vectors ξ ∈ T X,x , v ∈ E x . (f) (Strict) strong positivity means that at every point x ∈ X we can decompose Θ E,h as Θ E,h = j |α j ⊗ ψ j | 2 where Span(α j ⊗ ψ j ) = T * X,x ⊗ E * x . We will denote respectively by G , N , S (and > G , > N , > S ) the Griffiths, Nakano, strong (semi-)positivity relations. It is obvious that and one can show that the reverse implications do not hold when dim T > 1 and dim E > 1. The following result from [Dem80] will be useful. 5.5. Proposition. Let θ ∈ Herm(T ⊗ E), where (E, h) is a hermitian vector space. We define Tr E (θ) ∈ Herm(T ) to be the hermitian form such that where (e λ ) 1 λ r is an arbitrary orthonormal basis of E. Then As a consequence, if (E, h) is a Griffiths (semi-)positive vector bundle, then the tensor product (E ⊗ det E, h ⊗ det(h)) is strongly (semi-)positive.
Proof. Since [Dem80] is written in French and perhaps not so easy to find, we repeat here briefly the arguments. They are based on a Fourier inversion formula for discrete Fourier transforms. 5.6. Lemma. Let q be an integer 3, and x α , y β , 1 α, β r, be complex numbers. Let χ describe the set U r q of r-tuples of q-th roots of unity and put Then for every pair (λ, µ), 1 λ, µ r, the following identity holds: In fact, the coefficient of x α y β in the summation q −r χ∈U r q x(χ) y(χ) χ λ χ µ is given by so it is equal to 1 when the pairs {α, µ} and {β, λ} coincide, and is equal to 0 otherwise. The identity stated in Lemma 5.6 follows immediately. Now, let (t j ) 1 j n be a basis of T , (e λ ) 1 λ r an orthonormal basis of E and ξ = j ξ j t j ∈ T , w = j,λ w jλ t j ⊗ e λ ∈ T ⊗ E. The coefficients c jkλµ of θ with respect to the basis t j ⊗ e λ satisfy the symmetry relation c jkλµ = c kjµλ , and we have the formulas θ(w, w) = j,k,λ,µ c jkλµ w jλ w kµ , Tr E θ(ξ, ξ) = j,k,λ c jkλλ ξ j ξ k , (θ + Tr E θ ⊗ h)(w, w) = j,k,λ,µ c jkλµ w jλ w kµ + c jkλλ w jµ w kµ .

5.B. Chern form inequalities
In view of the estimates developed in section 6, we will have to evaluate integrals involving powers of curvature tensors, and the following basic inequalities will be useful. 5.8. Lemma. Let ℓ j ∈ (C r ) * , 1 j p, be non zero complex linear forms on C r , where (C r ) * ≃ C r is equipped with its standard hermitian form, and let µ the rotation invariant probability measure on S 2r−1 ⊂ C r . Then satisfies the following inequalities : Proof. Denote by dλ the Lebesgue measure on Euclidean space and by dσ the area measure of the sphere. One can easily check that the projection S 2r−1 → B 2r−2 , u = (u 1 , . . . , u r ) → v = (u 1 , . . . , u r−1 ), yields dσ(u) = dθ ∧ dλ(v) where u r = |u r | e iθ [ just check that the wedge products of both sides with 1 2 d|u| 2 are equal to dλ(u), and use the fact that dθ = 1 2i (du r /u r − du r /u r )], thus, in terms of polar coordinates v = t u ′ , u ′ ∈ S 2r−1 , we have dσ(u) = dθ ∧ t 2r−3 dt ∧ dσ ′ (u ′ ), and going back to the invariant probability measures µ on S 2r−1 and µ ′ on S 2r−3 , we get |u r | 2 = 1 − |v| 2 = 1 − t 2 and an equality If ℓ 1 , . . . , ℓ p are independent of u r , (5.9) and the Fubini theorem imply by homogeneity are then obtained by induction on r and p.
(a) For any ℓ ∈ (C r ) * , we can find orthonormal coordinates on C r such that ℓ(u) = |ℓ| u 1 in the new coordinates. Hence It follows from Hölder's inequality that and that the equality occurs if and only if all ℓ j are proportional.
(b, equality case) We argue by induction on r. For r = 1, we have in fact ℓ j (u) = α j u 1 , α j ∈ C * , and I(ℓ 1 , . . . , ℓ r ) = |ℓ j | 2 , thus the coefficient 1 (p+r−1)! = 1 p! is reached if and only if p 1. Now, assume r 2 and the equality case solved for dimension r − 1. By rescaling and reordering the ℓ j , we can always assume that ℓ j (e r ) = 0 (and hence ℓ j (e r ) = 1) for q + 1 j p, while ℓ j (e r ) = 0 for 1 j q (we can possibly have q = 0 here). Then we write ℓ j (u) = ℓ ′ j (u ′ ) for 1 j q and ℓ j (u) = ℓ ′ j (u ′ ) + u r for q + 1 j p. Therefore, if s k (ℓ ′ (u ′ )) denotes the k-th elementary symmetric function in (ℓ ′ j (u ′ ) q+1 j p , we find by what we have just proved. In an equivalent way, we get for all 0 q p − 1 and all choices of the forms ℓ ′ j ∈ (C r−1 ) * . In general, we can rotate coordinates in such a way that ℓ p (u) = u r and ℓ ′ p = 0, and we see that the above inequality holds when p is replaced by p − 1, as soon as q p − 2. Then the corresponding coefficients k = 0 for p, p − 1 are and since s 0 = 1, we infer that the inequality is strict. The only possibility for the equality case is q = p − 1, but then and we see that we must have equality in the case (r − 1, p − 1). By induction, we conclude that p − 1 r − 1 and that the ℓ j (u) = ℓ ′ j (u ′ ) are orthogonal for j p − 1, as desired.
5.12. Remark. When r = 2, our inequality (5.11) is equivalent to the "elementary" inequality relating a polynomial X p − s 1 X p−1 + · · · + (−1) p s p and its complex roots a j (just consider ℓ ′ j (u ′ ) = a j u 1 and ℓ j (u) = a j u 1 + u 2 on C 2 to get this). It should be observed that ( * ) is not optimal symptotically when p → +∞ ; in fact, Landau's inequality [Land05] gives max(1, |a j |) ( |s k | 2 ) 1/2 , from which one can easily derive that (1 + |a j | 2 ) 2 p |s k | 2 , which improves ( * ) as soon as p 7 (observe that 2 7 = 128 and k!(7 − k)! 3! 4! = 144). Our discussion of the equality case shows that inequality (5.8 (b)) is never sharp when p > r. It would be interesting, but probably challenging, if not impossible, to compute the optimal constant for all pairs (r, p), p > r, since this is an optimization problem involving the distribution of a large number of points in projective space.
We finally state one of the main consequences of these estimates concerning the Chern curvature form of a hermitian holomorphic vector bundle. 5.13. Proposition. Let T , E be complex vector spaces of respective dimensions dim T = n, dim E = r. Assume that E is equipped with a hermitian structure h, and denote by µ the unitary invariant probability measure µ on the unit sphere bundle S(E) = {u ∈ E ; |u| h = 1} of E.
as a pointwise weak inequality of (p − k, p − k)-forms. In particular, the above inequalities apply when (E, h) is a hermitian holomorphic vector bundle of rank r on a complex n-dimensional manifold X, and one takes θ j = Θ E,h to be the curvature tensor of E, so that Tr h θ j = c 1 (E, h) is the first Chern form of (E, h).

Proof. (a)
The assumption θ q S 0 means that at every point x ∈ X we can write θ as Without loss of generality, we can assume |ℓ qj | h * = 1. Then and since |ℓ qj | h * = 1, Lemma 5.8 (b) implies where is in the sense of the strong positivity of (p, p)-forms. The upper bound is obtained by the same argument, via 5.8 (a).
(b) By the definition of weak positivity of forms, it is enough to show the inequality in restriction to every (p − k)-dimensional subspace T ′ ⊂ T . Without loss of generality, we can assume that dim T = p − k (and then take T ′ = T ), that |ℓ j | = 1, and also that θ > G 0 (otherwise take a positive definite form η ∈ Λ 1,1 R T * , replace θ with θ ε = θ + ε η ⊗ h, and let ε tend to 0). For any u ∈ S(E), let 0 λ 1 (u) · · · λ p−k (u) be the eigenvalues of the hermitian form q u (•) = θ(u), u on T with respect to ω = Tr h θ = r j=1 θ(e j ), e j ∈ Herm(T ), ω > 0, (e j ) being any orthonormal frame of E. We have to show that However, the inequality between geometric and arithmetic means implies thus, putting Q(u) = 1 p−k Tr ω θ(u), u , Q ∈ Herm(E), it is enough to prove that (5.14) Our assumption θ > G 0 implies Q(u) = 1 j r c j |ℓ ′ qj (u)| 2 for some c j > 0 and some orthonormal basis (ℓ ′ qj ) 1 j r of E * , and Tr ω (ω) = 1.
5.15. Remark. For p = 1, the inequalities of Proposition 5.13 are identities, and no semi-positivity assumption is needed in that case. This can be seen directly from the fact that we have for every hermitian quadratic form Q on E. However, when p 2, inequality 5.13 (a) does not hold under the assumption that E G 0 (or even that E is dual Nakano semi-positive, i.e. E * Nakano semi-negative). Let us take for instance E = T P n ⊗ O(−1). It is well known that E is isomorphic to the tautological quotient vector bundle C n+1 /O(−1) over P n , and that its curvature tensor form for the Fubini-Study metric is given by The main qualitative result is summarized in the following statement.
By collecting all non negligible terms (6.13 1,2 ) and (6.14 1 ), we obtain a curvature form At this point, we come back to the orbifold situation, and thus replace σ by σ 1/ρ , ϕ by ρ −1 ϕ and ε by ρ 2 ε. This gives the curvature estimate In the general situation D = 1 j N (1 − 1/ρ j )∆ j of a multi-component orbifold divisor, we add the components ∆ j one by one, and obtain inductively the following quantitative estimate, which is a rephrasing of Theorem 0.9.
6.16. Corollary. With a choice of γ > γ V,D := max(max(d j /ρ j ), γ V ) 0 determined by the curvature assumptions of Proposition 6.1, and of hermitian metrics on A, V , O X (D) as prescribed by conditions (6.3 j ), the orbifold metric yields a curvature tensor θ V D ,γ,ε := γ Θ A,h A,δ ⊗Id−Θ V D ,h V D ,ε such that the associated quadratic form Q V D ,γ,ε on T X ⊗ V satisfies for ε N ≪ ε N −1 ≪ · · · ≪ ε 1 ≪ 1 the curvature estimate Here, the symbol ≃ means that the ratio of the left and right hand sides can be chosen in [1−α, 1+α] for any α > 0 prescribed in advance.

6.B. Evaluation of some Chern form integrals and their limits
Our aim is to apply Lemma 5.8 and Corollary 6.16 to compute Morse integrals of the curvature tensor of a directed orbifold (X, V, D), where D = j (1 − 1/ρ j )∆ j is transverse to V . Let A ∈ Pic(X) be an ample line bundle, and d j , γ V , γ > γ V,D be defined as in 6.16. We get hermitian metrics h V D ,ε on V D and corresponding curvature tensors θ V D ,γ,ε in C ∞ (X |D|, Λ 1,1 T * X ⊗ Hom(V, V )) that are "orbifold smooth", and such that θ V D ,γ,ε G 0. Given a smooth strongly positive (n − p, n − p)-form β S 0 on X, we want to evaluate the integrals where S ε (V D ) denotes the unit sphere bundle of V D with respect to h ε , and µ ε the unitary invariant probability measure on the sphere. Proposition 5.13 (b) and the Fubini theorem imply the upper bound (6.18) When β is closed, the upper bound can be evaluated by a cohomology class calculation, thanks to the following lemma.
6.19. Lemma. The (1, 1)-form Tr θ V D ,γ,ε 0 is closed and belongs to the cohomology class Proof. The trace can be seen as the curvature of with the determinant metric. Since all metrics have equivalent behaviour along |D| (and can be seen as orbifold smooth), Stokes' theorem shows that the cohomology class is independent of ε.
Formally, the result follows from (1.25). One can also consider the intersection product for all smooth closed (n − 1, n − 1)-forms β on X, and apply Corollary 6.16 (b) to evaluate the limit as ε → 0. This will be checked later as the special case p = 1 of (6.17).
We actually need even more general estimates. The proof follows again from the Fubini theorem.
6.20. Proposition. Consider orbifold directed structures (X, V, D s ), 1 s k, with D s = 1 j N (1− 1 ρ s,j )∆ j . We assume that the divisors D s are simple normal crossing divisors transverse to V , sharing the same components ∆ j . Let d j be the infimum of numbers λ ∈ R + such that λ A−∆ j is nef, and let γ V be the infimum of numbers γ 0 such that θ V,γ := γ Θ A,h A ⊗Id V −Θ V,h V G 0 for suitable hermitian metrics h V on V . Take p = (p 1 , . . . , p k ) ∈ N k such that p ′ = n−(p 1 +. . .+p k ) 0 and a smooth, closed, strongly positive (p ′ , p ′ ) form β S 0 on X. Then for every there exist hermitian metrics h V Ds ,εs on the orbifold vector bundles V D s such that When β is closed, we get a purely cohomological upper bound 6.21. Complement. When p 1 = . . . = p k = 1, formulas 6.20 (b) and 6.20 (c) are equalities.
Proof. This follows from Remark 5.15.
In general, getting a lower bound for I p,ε (β) and I k,p,ε (β) is substantially harder. We start with I p,ε (β) and content ourselves to evaluate the iterated limit (6.22) lim ε→0 I p,ε (β) := lim For this, we consider the expression of the curvature form in a neigborhood of an arbitrary point z 0 ∈ ∆ j 1 ∩ . . . ∩ ∆ jm (if z 0 ∈ X |∆|, we have m = 0). We take trivializations of the line bundles O X (∆ j ) so that the hermitian metrics have weights e −ϕ j with ϕ j (z 0 ) = dϕ j (z 0 ) = 0, and introduce the corresponding "orbifold" coordinates . . , j m , We complete these coordinates with n−m variables z ℓ that define coordinates along ∆ j 1 ∩. . .∩∆ jm . In this way, we get a n-tuple (t j,ε , z ℓ ) of complex numbers that provide local coordinates on the universal cover of Ω z 0 |D|, where Ω z 0 is a small neighborhood of z 0 . Viewed on X, the coordinates t j,ε are multivalued near z 0 , but we can make a "cut" in X along ∆ j to exclude the negligible set of points where σ j (z) ∈ R − , and take the argument in ] − π, π[, so that Arg(t j,ε ) ∈ ] − (1 − 1/ρ j )π, (1 − 1/ρ j )π[ . If we integrate over complex numbers t j,ε without such a restriction on the argument, the integral will have to be multiplied by the factor (1 − 1/ρ j ) to get the correct value. Since |σ j | is bounded, the range of the absolute value |t j,ε | is an interval ]O(ε 1/2 j ), +∞[ , thus t j,ε will cover asymptotically an entire angular sector in C as ε j → 0. In the above coordinates, we have since ∇ j σ j = dσ j − σ j ∂ϕ j and the weight ϕ j of the metric of O X (∆ j ) is smooth. Denote By Corollary 6.16, we have is a smooth (1, 0)-form near z 0 . The approximate equality ≃ in formula (6.27 1,2 ) involves the approximation |∇ j σ j (z)|/|∇ j σ j (z 0 )| ≃ 1, which holds in a sufficiently small neighborhood of z 0 ; if we apply the Fubini theorem and consider the fiber integral over z 0 ∈ X, there is actually no error coming from this approximation. Now, we want to integrate the volume form θ V D ,γ,ε · u, u p ∧ β dµ ε (u) along the fibers of S ε (V D ) → X. The sphere bundle S ε (V D ) is defined by For the sake of simplicity, we first deal with the case where the divisor D = (1 − 1/ρ j )∆ j has a single component. Along ∆ j , we then get an orthogonal decomposition V = (V ∩ T ∆ j ) ⊕ Ce j , and by (6.28) we can write We reparametrize the integration in u ∈ S ε (V D ) on the sphere S(V ) by introducing the change of variables so that u j,ε satisfies |u j,ε | 2 ε = |u| 2 and e * j (u j,ε ) = √ 1 − τ e * j (u) = 1 (1 + |t j,ε | 2 ) 1/2 e * j (u), |t j,ε | 2 |e * j (u j,ε )| 2 = τ |e * j (u)| 2 .
This gives dµ ε (u j,ε ) = dµ(u), and as a consequence (6.17) can be rewritten as (6.29) Finally, a use of polar coordinates with α = Arg(t j,ε ) shows that A substitution u → u j,ε in (6.27 1,2 ) yields The last term is a (1, 1)-form that is a square of a (1, 0)-form (when u is fixed), hence the expansion of the p-th power can involve at most one such factor. Therefore we get The integrals involving b j (u j,ε ) are of the form where A j,ε (u), A ′ j,ε (u) are forms with uniformly bounded coefficients in orbifold coordinates. Since |t j,ε | 2 1+|t j,ε | 2 is bounded by 1 and converges to 0 on X ∆ j , Lebesgue's dominated convergence theorem shows that the second integral converges to 0. The second integral can be estimated by the Cauchy-Schwarz inequality. We obtain an upper bound where the first factor converges to 0 and the second one is bounded by Fubini, since C i dt ∧ dt/(1 + |t| 2 ) 2 < +∞. Modulo negligible terms, and changing variables into our new parameters (τ, α), we finally obtain Since u j,ε → u almost everywhere and boundedly, we have Here, we have to remember that τ = τ j,ε converges uniformly to 0 (even in the C ∞ topology), on all compact subsets of X ∆ j , hence the second integral in (6.32) asymptotically concentrates on ∆ j as ε → 0. Also, the angle α = Arg(t j,ε ) runs over the interval ] − (1 − 1/ρ j )π, (1 − 1/ρ j )π[. In the easy case p = 1, we get If we assume β closed, this is equal to the intersection product 1 and the final assertion of the proof of Lemma 6.19 is thus confirmed, adding the components ∆ j one by one (see below). Now, in the general case p 1, we will obtain a lower bound of the second integral involving dτ ∧ dα in (6.32) by using a change of variable (1 − τ )|u ′ | 2 + |e * j (u)| 2 1/2 u. Since (1 − τ )|u ′ | 2 + |e * j (u)| 2 |u| 2 = 1, it is easy to check that dµ(h j,ε (u)) (1 − τ ) r−1 dµ(u) on the unit sphere, that |e * j (h j,ε (u))| |e * j (u)|, and finally, that Hence, by a change a variable u → h j,ε (u) we find Here, we have to remember that τ = τ j,ε converges uniformly to 0 (even in the C ∞ topology), on all compact subsets of X ∆ j . Therefore, the last integral concentrates over the divisor ∆ j . If we apply the binomial formula with an index q ′ = q − 1, we see that the limit as ε → 0 is equal to (1 − τ ) p−q+r−1 τ q−1 dτ ∧ β dµ(u). (1 − τ ) p−q+r−1 τ q−1 dτ = (p − q + r − 1)! (q − 1)! (p + r − 1)! and the combination of (6.29) and (6.32 − 6.35) implies Inductively, formula (6.36) requires the investigation of more general integrals (6.37) where Y is a subvariety of X (which we assume to be transverse to the ∆ j 's, and ℓ j ∈ C ∞ (Y, V * ) with |ℓ j | = 1, and β S 0 is a smooth form of suitable bidegree on Y . Not much is changed in the calculation, except that the change of variable u → g j,ε • h j,ε (u) applied to 1 j p ′ |ℓ j (u)| 2 introduces an extra factor (1 − τ ) p ′ in the lower bound, entirely compensated by the corresponding factor (1−τ ) p−p ′ −q appearing in θ V,γ,ε ·u, u p−p ′ . The binomial formula yields a coefficient p−p ′ −1 q−1 instead of p−1 q−1 . We thus obtain lim ε→0 I p,p ′ ,Y,ε (β) When D contains several components, we apply induction on N and put In this setting, (6.26) can be rewritten in the form of a decomposition By an iteration of our integral lower bound (6.38), we have to deal inductively with all intersections ∆ J = ∆ j 1 ∩ . . . ∩ ∆ jm , J = {j 1 , . . . , j m } ⊂ {1, . . . , N } ; we neglect the self-intersection terms, since when we take the limit as t j,ε → ∞. Those terms are equal to One would then have to evaluate the contribution of b j (u ′ j ), b j (u ′ j ) h j in the integral ∆ j .

Non probabilistic estimates of the Morse integrals
The non probabilistic estimate uses more explicit curvature inequalities and has the advantage of producing results also in the general orbifold case. Let us fix an ample line bundle A on X equipped with a smooth hermitian metric h A such that ω A := Θ A,h A > 0, and let γ V be the infimum of values λ ∈ R + such that in the sense of Griffiths. For any orbifold structure D = j (1 − 1/ρ j )∆ j , Corollary 6.16 then shows that the s-th directed orbifold bundle V s := V D (s) (cf. § 1.B) possesses hermitian metrics h V D (s) ,εs such that the associated curvature tensor satisfies the inequality provided we assume d j A − ∆ j nef and take In particular, any value is admissible, and we can apply the estimates 6.41 (b) and (6.42) with these values. Instead of exploiting a Monte Carlo convergence process for the curvature tensor as was done in § 4.B, we are going to use a more precise lower bound of the curvature tensor Θ L τ,k ,ε of the orbifold rank 1 sheaf associated with F = τ A, τ ≪ 1, namely Our formulas 3.20 (a,b) become Under the assumption (7.3 ′ ), we have g k,γ,ε (z, x, u) 0, but in general this is not true for g k,0,ε (z, x, u), so we express g k,0,ε (z, x, u) as a difference of g k,γ,ε (z, x, u) and of a multiple of ω A . By definition θ s,γ,ε = γ s ω A ⊗ Id + θ s,0,ε , and we get Θ L τ,k ,ε = ω r,k,b + α ε − β, where (7.6) (7.14 1 ) A simpler (but larger) upper bound is (7.14 2 ) Finally, inequality (7.7) translates into (7.15) 1 (n + kr − 1)! X k (V D )(L τ,k , 1) Θ n+kr−1 L τ,k ,ε 1 n! k! r (kr − 1)! (M n,k − M ′ n,k ).
If we put everything together, we get the following (complicated!) existence criterion for orbifold jet differentials.
7.16. Existence criterion. Let (X, V, D) with D = 1 j N (1 − 1/ρ j )∆ j be a directed orbifold, and let A be an ample line bundle on X. Assume that D is a simple normal crossing divisor transverse to V , that c 1 (∆ j ) = d j c 1 (A), c 1 (V * ) = λ V c 1 (A) and let γ V be the infimum of values γ > 0 such that Then, a sufficient condition for the existence of (many) non zero holomorphic sections of multiples of where M n,k admits the lower bounds (7.10 2 ) or (7.12), and M ′ n,k admits the upper bound (7.14 2 ).

7.B. Compact case (no boundary divisor)
We address here the case of a compact (projective) directed manifold (X, V ), with a boundary divisor D = 0. By (7.10 2 ) and (7.14 2 ), we find Therefore, for τ > 0 sufficiently small, M n,k − M ′ n,k is positive as soon as k n and ( 7.18. Example. In the case where X is a smooth hypersurface of P n+1 of degree d and V = T X , we have r = n and det(V * ) = O(d − n − 2). We take A = O(1). If Q is the tautological quotient bundle on P n+1 , it is well known that T P n+1 ≃ Q ⊗ O(1) and det Q = O(1), hence T * P n+1 ⊗ O(2) = Q * ⊗ O(1) = Λ n Q G 0, and the surjective morphism implies that we also have V * ⊗ O(2) G 0. Therefore, we find γ V = 2 and λ V = d − n − 2. The above condition (7.17) becomes k n and k n and d > 2 n!
This lower bound improves the one stated in [Dem12], but is unfortunately far from being optimal. Better bounds -still probably non optimal -have been obtained in [Dar16] and [MTa19].

7.C. Logarithmic case
The logarithmic situation makes essentially no difference in treatment with the compact case, except for the fact that we have to replace V by the logarithmic directed structure V D , and the numbers γ V , λ V by We get the sufficient condition (7.20) k n and λ V D > n!
For X = P n , V = T P n , and for a divisor D = ∆ j of total degree d on P n , we can still take γ V D = 2 by Proposition 5.8, and we have det(V * D ) = O(d − n − 1). We get the degree condition (7.21) k n and d > 2 n! 1 s k 1 s n − n + 1.
In fact, c 1 = 1, c 2 = 32.5 and c n n 5 for all n ∈ N * , hence the above requirement implies in any case the inequality n 2 t 1 n 3 d j . The Stirling and Euler-Maclaurin formulas give (7.26 ′ ) c n ∼ (2π) 1/2 n n+7/2 e −n (γ + log n) n−1 as n → +∞, where γ = 0.577215 . . . is the Euler constant, the ratio being actually bounded above for n 3 by exp (1/2)(1 − 1/n)/(γ + log n) + 13/12n − 1/n 2 → 1. Let us observe that In this way, we get the sufficient condition For instance, if we take all components ∆ j possessing the same degrees d j = d and ramification number ρ j ρ, these numbers and the number N of components have to satisfy the sufficient condition This possibly allows a single component (taking d, ρ large), or d, ρ small (taking N large). Since we have neglected many terms in the above calculations, the "technological constant" c n appearing in these estimates is probably much larger than needed. Notice that the above estimates require jets of order k n and ramification numbers ρ > n. Parts (a) and (a ′ ) of Theorem 0.8 follow from (7.27) and (7.27 N ).
Parts (b) and (b ′ ) of Theorem 0.8 follow from (7.29) and (7.29 N ). Again, the constant 2 n−1 (2n − 1) n n is certainly far from being optimal. Answering the problem raised in Remark 6.43 might help to improve the bounds.

Appendix: a proof of the orbifold vanishing theorem
The orbifold vanishing theorem is proved in [CDR20] in the case of boundary divisors D = (1 − 1/ρ j )∆ j with rational multiplicities ρ j ∈ ]1, ∞]. However, the definition of orbifold curves shows that we can replace ρ j by ⌈ρ j ⌉ ∈ N ∪ {∞} without modifying the space of curves we have to deal with. On the other hand, this replacement makes the corresponding sheaves E k,m V * D larger. Therefore, the case of arbitrary real multiplicities ρ j ∈ ]1, ∞] stated in Proposition 0.7 follows from the case of integer multiplicities. We sketch here an alternative and possibly more direct proof of Proposition 0.7, by checking that we can still apply the Ahlfors-Schwarz lemma argument of [Dem97] in the orbifold context. For this, we associate to D the "logarithmic divisor" and, assuming (X, V, D ′ ) non singular, we make use of the tower of logarithmic Semple bundles (8.1) X S k (V D ′ ) → X S k−1 (V D ′ ) → · · · → X S 1 (V D ′ ) → X S 0 (V D ′ ) := X (in reference to the work of the British mathematician John Greenlees Semple, see [Sem54]), where each stage is a smooth directed manifold (X S k (V D ′ ), V k D ′ ) defined inductively by (8.2) X S k (V D ′ ) := P (V k−1 D ′ ) = projective bundle of lines of V k−1 D ′ , and V k D ′ is a subbundle of the logarithmic tangent bundle of X S k (V D ′ ) associated with the pull-back of D ′ . Each of these projective bundles is equipped with a tautological line bundle O X S k (V D ′ ) (−1) (see [Dem97] for details), and V k D ′ consists of the elements of the logarithmic tangent bundle that project onto the tautological line, so that we have an exact sequence be the natural projection. Then the top-down projection π k,0 : X S k (V D ′ ) → X yields a direct image sheaf (8.3) (π k,0 ) * O X S k (V D ′ ) (m) := E S k,m V * D ′ ⊂ E k,m V * D ′ . Its stalk at point x ∈ X consists of the algebraic differential operators P (f [k] ) acting on germs of k-jets f : (C, 0) → (X, x) tangent to V , satisfying the invariance property whenever ϕ ∈ G k is in the group of k-jets of biholomorphisms ϕ : (C, 0) → (C, 0). By construction, the sheaf of orbifold jet differentials E k,m V * D is contained in E k,m V * D ′ , and we have a corresponding inclusion (8.5) E S k,m V * D ⊂ E S k,m V * D ′ of the Semple orbifold jet differentials into the Semple logarithmic differentials. A consideration of the algebra E S k,m V * D makes clear that there exists a submultiplicative sequence of ideal sheaves (J D,k,m ) m∈N on X S k (V D ′ ), such that the image of π * It is clear that the zero variety of V (J D,k,m ) projects into the support |D ′ | = |D| of D. We consider a smooth log resolution (8.7) µ k :X k → X S k (V D ′ ) of the ideal J D,k,m in X S k (V D ′ ), so that µ * k (J D,k,m ) = OX k (−G D,k,m ) for a suitable effective simple normal crossing divisor G D,k,m onX k that projects into |D| in X.
is very ample onX k . Finally, we select c ∈ N * such that (8.15) O X (cA − D ′ ) is very ample on X.
By taking the tensor product of (8.12 − 8.15), (8.15) being raised to a power t ∈ N * , we find that (8.16) L k,m := OX k (m) ⊗ OX k (a) ⊗ OX k (−G D,k,m − H D,k ) ⊗π * k,0 O X ((s + b + tc)A − tD ′ ) is very ample onX k . We will later need to take t = |a| = ℓ a ℓ , which is of course an admissible choice.
8.17. Lemma. Let (X, V, D) be a projective non singular directed orbifold, and A an ample divisor on X. Then, for every orbifold entire curve f : C → (X, V, D) and every section . . , f (k) ) = 0.
Proof. As we have already seen for local sections, every global jet differential P in H 0 X, E S k,m V * D ⊗ O X (−A) gives rise to sections ). Assume that P (f [k] ) = 0 (so that, in particularσ P = 0). We consider a basis (g j ) of sections of L k,m in (8.16), the canonical section η D,k ∈ H 0 (X k , OX k (H D,k )) and take the products (8.18) h j = g j (σ P ) q−1 (τ D ′ ) t η D,k ∈ H 0 X k , OX k (mq) ⊗ OX k (a) ⊗ OX k (−qG D,k,m ) where q = s + b + tc + 1. We now observe, thanks to our choice t = |a| = a ℓ , that h j (f [k] ) · (f ′ k−1 ) mq · 1 ℓ k (dπ k,ℓ (f ′ k−1 )) a ℓ (8.19) is a product of holomorphic sections on C, by (8.11) and (8.9˜) combined with (8.16) and (8.18), and the fact that P (f [k] ) =σ P (f [k] ) · (f ′ k−1 ) m is holomorphic with values in f * O X (−A). The product also takes value in the trivial bundle over C, and can thus be seen as a holomorphic function. As j varies, these functions are not all equal to zero, and we define a hermitian metric γ(t) = γ 0 (t) |dt| 2 on the complex line C by putting where ψ is a quasi plurisubharmonic potential onX k which will be chosen later. Notice that γ 0 (t) is locally bounded from above and almost everywhere non zero. Since (8.19) only involves holomorphic factors in the right hand side, we get (8.21) i ∂∂ log γ 0 1 mq + |a| (f [k] ) * (ω k + i ∂∂ψ) whereω k = i ∂∂ log |g j | 2 is a Kähler metric onX k , equal to the curvature of the very ample line bundle L k,m for the projective embedding provided by (g j ). (In fact, (8.21) could be turned into an equality by adding a suitable sum of Dirac masses in the right hand side). Of course, ψ will be taken to be an ω-plurisuharmonic potential onX k . We wish to get a contradiction by means of the Ahlfors-Schwarz lemma (see e.g. [Dem97, Lemma 3.2]), by showing that i ∂∂ log γ 0 Aγ for some A > 0, an impossibility for a hermitian metric on the entire complex line. Since ψ is locally bounded from above, by (8.19) and the inequality between geometric and arithmetic means, we have where C > 0 and the norms |h j | 2 and |f ′ [k−1] (t)| 2 log are computed with respect to smooth metrics on OX k (mq)⊗OX k (a)⊗OX k (−qG D,k,m ) and on the logarithmic tautological line bundle O X S k (V D ′ ) (−1), respectively. The term |h j | 2 is bounded, but one has to pay attention to the fact that |f ′ [k−1] (t)| 2 log has poles on f −1 (|D ′ |). If we use local coordinates (z 1 , . . . , z n ) on X such that ∆ j = {z j = 0}, we have in terms of a smooth Kähler metric ω k−1 on X S k (V D ′ ). What saves us is that h j contains a factor τ D ′ (f ) t that vanishes along all components ∆ j . Therefore (8.22) implies the existence of a number δ > 0 such that Since the morphismπ k,k−1 has a bounded differential and f ′ [k−1] (t) = dπ k,k−1 (f ′ [k] (t)), we infer By (8.21) and (8.22 ′′ ), in order to get a lower bound i ∂∂ log γ 0 Aγ, we only need to choose the potential ψ so that (8.23) j |f j | −2+2δ |f ′ j | 2 ≤ C ′′′ (f [k] ) * (ω k + i ∂∂ψ).
If τ j ∈ H 0 (X, O X (∆ j )) is the canonical section of divisor ∆ j , (8.23) is achieved by taking ψ = ε j |τ j •π k,0 | 2δ , for any choice of a smooth hermitian metric on O X (∆ j ) and ε > 0 small enough. In some sense, we have to take a suitable orbifold Kähler metricω k + i ∂∂ψ onX k to be able to apply the Ahlfors-Schwarz lemma. It might be interesting to find the optimal choice of δ > 0, but this is not needed in our proof.
End of the proof of Proposition 0.7. We still have to extend the vanishing result to the case of non necessarily G k -invariant orbifold jet differentials P ∈ H 0 (X, E k,m V * D ⊗ O X (−A)).
In particular deg P α < m unless α = (m, 0, . . . , 0), in which case P α = P . If the result is known for degrees < m, then all P α (f [k] ) vanish for P α = P and one can reduce the proof to the invariant case by induction, as the term P α of minimal degree is invariant. The proof makes use of induced directed structures, and is purely formal and group theoretic. Essentially, the argument is that P becomes an invariant jet differential when restricted to the subvariety of the Semple k-jet bundle consisting of germs g [k] of k-jets such that P α (g [k] ) = 0 for P α = P . Singularities may appear in this subvariety, but this does not affect the proof since the induced directed structure is embedded in the non singular logarithmic Semple tower. We refer the reader to [Dem20, § 7.E] and [Dem20, Theorem 8.15] for details.